Before proceeding with our main results, let us remark that it is often easier to verify BJ orthogonality with the help of James' criteria [1]} by which \(x\perp y\)  if and only if there exists a supporting functional \(f_x\)  at \(x\)  (i.e., a norm-one linear functional \(f_x\colon X\rightarrow \mathbb {F}\)  with \(f_x(x)=\Vert x\Vert \)  which annihilates \(y\) ). One should mention here that James in [1]} assumed real normed spaces but the proof works also for the complex ones (the necessity part of the proof only requires the fact that if unit vectors satisfy \(x\perp y\) , then a linear functional \(f\)  which maps \(x\)  to 1 and annihilates \(y\)  is norm-one on the span of \(x,y\) ; using the Hahn–Banach theorem one then extends the domain of this \(f\)  from \(\mathop {\mathrm {span}}_{\mathbb {C}}\lbrace x,y\rbrace \)  to \(X\) ). We refer also to Kittaneh [3]} for a shorter proof in real or complex spaces in case the norm is smooth at \(x\) .
